Solutions of nonlinear PDEs in the sense of averages

DOI10.1016/j.matpur.2011.07.001zbMath1236.35083OpenAlexW2035090944WikidataQ110125247 ScholiaQ110125247MaRDI QIDQ662023

Mikko Parviainen, Juan J. Manfredi, Bernhard Kawohl

Publication date: 11 February 2012

Published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.matpur.2011.07.001

zbMATH Keywords

asymptotic expansionmean value theoremstochastic gamesviscosity solutionsdynamic programming principle\(p\)-harmonictwo-player zero-sum games

Mathematics Subject Classification ID

Integral representations of solutions to PDEs (35C15) Stochastic games, stochastic differential games (91A15) Asymptotic expansions of solutions to PDEs (35C20) Boundary behavior of harmonic functions in higher dimensions (31B25) Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games (49L25) Viscosity solutions to PDEs (35D40) Quasilinear elliptic equations with (p)-Laplacian (35J92)

Related Items

Convergence of dynamic programming principles for the \(p\)-Laplacian ⋮ Parabolic biased infinity Laplacian equation related to the biased tug-of-war ⋮ Asymptotic mean value properties for fractional anisotropic operators ⋮ An asymptotic expansion for the fractional p-Laplacian and for gradient-dependent nonlocal operators ⋮ An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term ⋮ \(C^{1,\alpha}\) regularity for the normalized \(p\)-Poisson problem ⋮ Regularity for nonlinear stochastic games ⋮ A natural approach to the asymptotic mean value property for the \(p\)-Laplacian ⋮ On the Aleksandrov-Bakelman-Pucci estimate for the infinity Laplacian ⋮ Random walks and random tug of war in the Heisenberg group ⋮ Variational approach to the asymptotic mean-value property for the \(p\)-Laplacian on Carnot groups ⋮ Gradient walks and $p$-harmonic functions ⋮ How I met the normalized \(p\)-Laplacian \(\Delta_p^N\) and what we know now about mean values and concavity properties ⋮ A game theoretical approximation for a parabolic/elliptic system with different operators ⋮ Asymptotic average solutions to linear second order semi-elliptic PDEs: a Pizzetti-type theorem ⋮ A game theoretical approximation for solutions to nonlinear systems with obstacle-type equations ⋮ A game theoretical approach for a nonlinear system driven by elliptic operators ⋮ Potential theory for the sum of the \(1\)-Laplacian and \(p\)-Laplacian ⋮ Time-dependent tug-of-war games and normalized parabolic \(p\)-Laplace equations ⋮ Tug-of-war games with varying probabilities and the normalized \(p(x)\)-Laplacian ⋮ Remarks on regularity for \(p\)-Laplacian type equations in non-divergence form ⋮ Maximal operators for the \(p\)-Laplacian family ⋮ On the geometry of the \(p\)-Laplacian operator ⋮ Up-to boundary regularity for a singular perturbation problem of p-Laplacian type equations in non-divergence form ⋮ A mean value formula for the variational \(p\)-Laplacian ⋮ Comparison principles for infinity-Laplace equations in Finsler metrics ⋮ Characterization by asymptotic mean formulas of \(q\)-harmonic functions in Carnot groups ⋮ Harnack inequality and an asymptotic mean-value property for the Finsler infinity-Laplacian ⋮ Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle ⋮ A continuous time tug-of-war game for parabolic p(x,t)-Laplace-type equations ⋮ A finite difference method for the variational \(p\)-Laplacian ⋮ Convergence of the naturalp-means for thep-Laplacian ⋮ Gradient and Lipschitz Estimates for Tug-of-War Type Games ⋮ Tug-of-war with Kolmogorov ⋮ Mean value properties of fractional second order operators ⋮ Mean value properties and unique continuation ⋮ Statistical exponential formulas for homogeneous diffusion ⋮ Viscosity solutions to a parabolic inhomogeneous equation associated with infinity Laplacian

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• An evolution equation involving the normalized \(p\)-Laplacian
• The maximum principle for semicontinuous functions
• Motion of level sets by mean curvature. I
• Tug-of-war with noise: a game-theoretic view of the \(p\)-Laplacian
• Strong maximum principle for semicontinuous viscosity solutions of nonlinear partial differential equations
• Generalized motion by curvature with a Dirichlet condition
• Subharmonic functions in sub-Riemannian settings
• On absolutely minimizing Lipschitz extensions and PDE \(\bigtriangleup_{\infty}(u)=0\)
• On the evolution governed by the infinity Laplacian
• On the Equivalence of Viscosity Solutions and Weak Solutions for a Quasi-Linear Equation
• On the definition and properties of $p$-harmonious functions
• An Asymptotic Mean Value Characterization for a Class of Nonlinear Parabolic Equations Related to Tug-of-War Games
• Tug-of-war and the infinity Laplacian
• Stratified Lie Groups and Potential Theory for their Sub-Laplacians
• A PDE Perspective of the Normalized Infinity Laplacian
• User’s guide to viscosity solutions of second order partial differential equations
• Harmonious Extensions
• p-Harmonic approximation of functions of least gradient
• Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations

This page was built for publication: Solutions of nonlinear PDEs in the sense of averages